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Lesson Directive // Exponents & RootsREF_CORE

What Is a Power?

aan^n==aa×\timesaa×\times⋯\cdots

Hover over a variable in the formula above, or see glossary below:

aa
Base
Real Number
nn
Exponent / Power
Integer

a^n means the base a is multiplied by itself n times. For example, 2^4 = 2 \times 2 \times 2 \times 2 = 16.

INSIGHT: Exponentiation is repeated multiplication, just as multiplication is repeated addition.

Key Exponent Laws

a^m \times a^n = a^{m+n} (same base, add exponents). (a^m)^n = a^{mn} (power of a power). a^0 = 1 for any a \ne 0.

INSIGHT: The laws of exponents let you simplify complex expressions without computing huge numbers.

Square & Cube Roots

The square root \sqrt{a} is the inverse of squaring: \sqrt{9} = 3 because 3^2 = 9. Roots are fractional exponents: \sqrt{a} = a^{1/2}.

INSIGHT: $\sqrt{a} = a^{1/2}$ — roots are just fractional exponents.
Detailed Theory & ReferencesEXT_DOC

Exponents and Radicals

Exponents (Powers)

an=a×a×⋯×a⏟n factorsa^n = \underbrace{a \times a \times \cdots \times a}_{n \text{ factors}}

Laws of Exponents

LawFormulaExample
Product Ruleam⋅an=am+na^m \cdot a^n = a^{m+n}23⋅24=272^3 \cdot 2^4 = 2^7
Quotient Ruleaman=am−n\dfrac{a^m}{a^n} = a^{m-n}5652=54\dfrac{5^6}{5^2} = 5^4
Power of a Power(am)n=amn(a^m)^n = a^{mn}(32)3=36(3^2)^3 = 3^6
Zero Exponenta0=1a^0 = 170=17^0 = 1
Negative Exponenta−n=1ana^{-n} = \dfrac{1}{a^n}2−3=182^{-3} = \dfrac{1}{8}

Radicals (Roots)

an=a1/n\sqrt[n]{a} = a^{1/n}

Scientific Notation

93,000,000=9.3×10793{,}000{,}000 = 9.3 \times 10^7

References

AI NOTICE

AI Assistance Disclaimer: This module uses AI-assisted educational models and interactive visual representations to help explain scientific and mathematical concepts. For formal research or academic evaluation, please verify formulas and data against standard primary reference materials.